The Outer Regularity of the Hewitt-Stromberg Measures in a Metric Space and Applications

The aim of this paper is to show that if the Hewitt-Stromberg pre-measures are finite, then these pre-measures have a kind of outer regularity in a general metric space X. We also give some conditions on the Hewitt-Stromberg pre-measures such that the Hewitt-Stromberg measures are regular on a complete separable metric space X. In addition, if the measure µ satisfies the doubling condition then the Hewitt-Stromberg pre-measure and measure are both zero or nonzero. As an application, we obtain new sufficient conditions for the valid refined multifractal formalism which is based on the Hewitt-Stromberg measures.